This entry contributed by Nicolas Bray

In general, a graph product of two graphs G and H is a new graph whose vertex set is and where, for any two vertices (g, h) and in the product, the adjacency of those two vertices is determined entirely by the adjacency (or equality, or non-adjacency) of g and , and that of h and . There are cases to be decided (three possibilities for each, with the case where both are equal eliminated) and thus there are different types of graph products that can be defined.

The most commonly used graph products, given by conditions sufficient and necessary for adjacency, are summarized in the following table (Hartnell and Rall 1998). Note that the terminology is not quite standardized, so these products may actually be referred to by different names by different sources (for example, the graph lexicographic product is also known as the graph composition; Harary 1994, p 21). Many other graph products can be found in Jensen and Toft (1994).

graph product name	symbol	definition
graph Cartesian Product		( and ) or ( and )
graph categorical Product		( and )
graph lexicographic product		( ) or ( and )
graph strong product		( and ) or ( and ) or ( and )

Graph Cartesian Product, Graph Categorical Product, Graph Composition, Graph Lexicographic Product, Graph Strong Product, Graph Sum

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